Inverse Trigonometric and Hyperbolic Functions

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• Misc
  • Math colloquia
    • Leary, “Axioms, Models, Infinity, and the Nature of the Universe”
      • Thursday, Feb. 5, 2:30 - 3:30, Newton 203
    • Dickerson, “Mathematics Majors’ Understandings of the Concept of Mathematical Definition”
      • Monday, Feb. 9 4:00 - 5:00, Newton 201
        • (Time is corrected from what was said in class)
    • Extra credit for going and writing summary/reaction to either or both
  • Math Olympiad
    • Regional college math contest
    • Saturday, Feb. 7, 9:30 - 12:30
    • See Prof. Towsley for more info or to sign up
• Questions?
  • Problem set question 1
    • Solve e^{-0.01 t} = 1000
    • Isolate t
• Inverse Trigonometric and Hyperbolic Functions
  • Section 7.6 and “Definitions” subsection of 7.7
    • sin^-1 ranges from -π/2 to π/2
    • cos^-1 ranges from 0 to π
    • tan^-1 ranges from -π/2 to π/2
    • Vertical asymptotes become horizontal asymptotes of inverse functions
    • Notation
      • sinⁿx usually means (sin x)ⁿ
      • But not if n = -1
      • sin^-1x means function that inverts sinx
    • Identities
      • sin^-1x + cos^-1x = π/2
      • Similarly for tan/cot, sec/csc
    • Derivatives generally of form 1 / …
      
      • “co-” functions’ derivatives are negatives of these
      • All derived using derivative-of-inverse rule from section 7.1
    • Anti-derivatives of various forms 1 / …
    • Hyperbolic functions
      • “Hyperbolic” refers to these functions locating points along a hyperbola analogous to how ordinary trig functions locate points around a circle
        
  • Examples
    • Suppose the attic of my house is about 30 feet across with a 12-foot high peaked roof over it. What angle does the roof make with the floor?
      
    • What is sinh 0?
      • Definition: sinh x = ( e^x - e^-x ) / 2
      • So sinh 0 = ( e⁰ - e⁰ ) / 2 = 0
    • Differentiate sec^-1(x+1)
      
    • Differentiate 3 tan^-1(x²)
      
    • Integrate 1 / (1+4x²)
      
• Next
  • L’Hôpital’s Rule
  • Section 7.5

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